9.1 Supersonic flow

Supersonic flow occurs when air moves faster than the speed of sound, creating phenomena like shock waves and expansion waves that don't exist at lower speeds. These effects cause abrupt changes in pressure, temperature, and density around objects, which fundamentally changes how engineers must approach aircraft and propulsion system design.

This guide covers the core physics of supersonic flow: Mach number classification, shock and expansion wave behavior, the governing conservation equations, nozzle flow, and supersonic airfoil theory.

Supersonic flow characteristics

Supersonic flow happens when the flow velocity exceeds the local speed of sound. At these speeds, the air can no longer "get out of the way" ahead of the object, because pressure disturbances (which travel at the speed of sound) can't propagate upstream. This is why compressibility effects dominate and shock waves form.

The Mach number is the single most important parameter for characterizing these flows. It tells you the ratio of flow speed to the local speed of sound, and it determines which physical phenomena you need to account for.

Mach number vs flow speed

The Mach number ($M$) is defined as:

$M = \frac{V}{a}$

where $V$ is the flow speed and $a$ is the local speed of sound.

Flow regimes based on Mach number:

• Subsonic: $M < 1$
• Transonic: $M \approx 1$ (roughly 0.8 to 1.2)
• Supersonic: $M > 1$
• Hypersonic: $M > 5$ (typically)

As the Mach number increases, compressibility effects grow more significant. Density, pressure, and temperature all change substantially, and you can no longer treat air as incompressible the way you can at low subsonic speeds.

Shock waves in supersonic flow

Shock waves are extremely thin regions (on the order of a few mean free paths) where flow properties change abruptly. They form when supersonic flow is forced to decelerate.

• Normal shocks are perpendicular to the flow direction.
• Oblique shocks are inclined at an angle to the flow direction.

Across any shock wave, pressure, density, and temperature all increase, while velocity decreases. The process is irreversible, meaning entropy increases and total pressure drops.

Expansion waves in supersonic flow

Expansion waves are the opposite of shock waves. They occur when supersonic flow turns around a convex corner or expands into a lower-pressure region.

• The flow accelerates and the Mach number increases.
• Pressure, density, and temperature all decrease.
• The process is isentropic (reversible and adiabatic), so there's no loss in total pressure.

Prandtl-Meyer expansion waves are the specific type that occurs at sharp convex corners in supersonic flow. They're a collection of Mach waves that fan out from the corner.

Compressibility effects on density

In supersonic flow, density changes are large and cannot be ignored. The density ratio across a normal shock wave is:

$\frac{\rho_2}{\rho_1} = \frac{(\gamma+1)M_1^2}{(\gamma-1)M_1^2+2}$

where $\rho_1$ and $\rho_2$ are upstream and downstream densities, $M_1$ is the upstream Mach number, and $\gamma$ is the specific heat ratio (1.4 for air at standard conditions).

Density increases across shock waves and decreases across expansion waves. Notice that as $M_1 \to \infty$, the density ratio approaches a finite limit of $\frac{\gamma+1}{\gamma-1}$ (which equals 6 for air). This means there's a maximum compression ratio a normal shock can produce, no matter how fast the flow.

Temperature changes across shocks

The temperature ratio across a normal shock is:

$\frac{T_2}{T_1} = \frac{[2\gamma M_1^2-(\gamma-1)][(\gamma-1)M_1^2+2]}{(\gamma+1)^2M_1^2}$

Unlike the density ratio, the temperature ratio has no finite upper limit. At high Mach numbers, temperatures behind shock waves can reach thousands of degrees. For example, at Mach 3 in standard atmosphere conditions, the temperature behind a normal shock roughly doubles. This is a major driver of material selection and thermal protection system design on supersonic and hypersonic vehicles.

Governing equations of supersonic flow

The governing equations come from the three fundamental conservation laws (mass, momentum, energy) plus the equation of state. Together, they form a closed system that lets you solve for all flow properties across shocks, through nozzles, and around bodies.

Continuity equation for compressible flow

The continuity equation enforces conservation of mass. For steady, one-dimensional flow:

$\rho_1 A_1 V_1 = \rho_2 A_2 V_2$

where $A$ is the cross-sectional area. Unlike incompressible flow (where $\rho$ is constant and you only worry about $A$ and $V$), here all three quantities can change simultaneously. This is what makes compressible flow analysis more involved.

Momentum equation for compressible flow

The momentum equation enforces conservation of momentum. For steady, one-dimensional flow:

$p_1 + \rho_1 V_1^2 = p_2 + \rho_2 V_2^2$

This relates changes in pressure to changes in density and velocity. The $\rho V^2$ term represents the momentum flux, and it becomes dominant at high Mach numbers.

Energy equation for compressible flow

The energy equation enforces conservation of energy. For steady, one-dimensional, adiabatic flow with no work:

$h_1 + \frac{V_1^2}{2} = h_2 + \frac{V_2^2}{2}$

where $h$ is the specific enthalpy. The sum $h + \frac{V^2}{2}$ is the stagnation enthalpy, and it remains constant along a streamline in adiabatic flow. This means that as kinetic energy increases (flow speeds up), enthalpy (and therefore temperature) must decrease, and vice versa.

Equation of state for ideal gases

The ideal gas equation of state closes the system:

$p = \rho R T$

where $R$ is the specific gas constant (287 J/(kg·K) for air). Combined with the three conservation equations, this gives you four equations for four unknowns ($p$, $\rho$, $T$, $V$) at any station in the flow.

Normal shock waves

Normal shock waves are perpendicular to the flow direction. They represent the simplest type of shock and are the foundation for understanding more complex shock configurations. They occur when supersonic flow encounters a blunt obstruction or when conditions in a nozzle force a sudden deceleration.

Normal shock wave properties

Three key facts about normal shocks:

• Pressure, density, and temperature all increase across the shock, while velocity decreases.
• The downstream flow is always subsonic ($M_2 < 1$), regardless of how high $M_1$ is.
• Entropy increases across the shock, so the process is irreversible. Total (stagnation) pressure always drops.

The stronger the shock (higher $M_1$), the greater the losses. This is why supersonic vehicle designers try to avoid strong normal shocks whenever possible.

Rankine-Hugoniot equations

The Rankine-Hugoniot equations are the set of relations that describe property changes across a normal shock. They're derived by applying the continuity, momentum, and energy equations (plus the equation of state) across the shock. These relations connect upstream conditions (subscript 1) to downstream conditions (subscript 2) and are the basis for all the ratio formulas below.

Mach number relations across normal shocks

The downstream Mach number is related to the upstream Mach number by:

$M_2^2 = \frac{1+\frac{\gamma-1}{2}M_1^2}{\gamma M_1^2-\frac{\gamma-1}{2}}$

For a given $M_1 > 1$, this always gives $M_2 < 1$. As $M_1 \to \infty$, the downstream Mach number approaches a limiting value:

$M_2 \to \sqrt{\frac{\gamma-1}{2\gamma}}$

For air ($\gamma = 1.4$), this limit is approximately 0.378.

Pressure ratio vs Mach number

$\frac{p_2}{p_1} = 1+\frac{2\gamma}{\gamma+1}(M_1^2-1)$

This grows without bound as $M_1$ increases. For example, at $M_1 = 2$ with $\gamma = 1.4$, the pressure ratio is about 4.5. At $M_1 = 3$, it jumps to about 10.33.

Temperature ratio vs Mach number

$\frac{T_2}{T_1} = \frac{[2\gamma M_1^2-(\gamma-1)][(\gamma-1)M_1^2+2]}{(\gamma+1)^2M_1^2}$

The temperature ratio also increases without bound as $M_1$ increases. This is why aerodynamic heating is such a critical concern at high Mach numbers.

Density ratio vs Mach number

$\frac{\rho_2}{\rho_1} = \frac{(\gamma+1)M_1^2}{(\gamma-1)M_1^2+2}$

Unlike pressure and temperature, the density ratio has a finite upper limit of $\frac{\gamma+1}{\gamma-1} = 6$ for air. Even an infinitely strong normal shock can only compress air by a factor of 6.

Oblique shock waves

Oblique shock waves are inclined at an angle to the incoming flow. They're more common in practice than normal shocks because they form whenever supersonic flow encounters a wedge, a ramp, or any surface that deflects the flow. They're also weaker than normal shocks for the same upstream Mach number, which means lower losses.

Oblique shock wave geometry

Two angles define an oblique shock:

• Shock angle ($\beta$): The angle between the shock wave and the upstream flow direction.
• Deflection angle ($\theta$): The angle through which the flow turns after passing through the shock.

A normal shock is just a special case where $\beta = 90°$ and $\theta = 0°$.

Oblique shock wave properties

• Pressure, density, and temperature increase across the shock, while the velocity component normal to the shock decreases.
• Unlike normal shocks, the downstream flow can be either subsonic or supersonic, depending on $M_1$ and $\beta$.
• Entropy increases (irreversible process), but the losses are generally smaller than for a normal shock at the same $M_1$.

Oblique shock wave equations

The oblique shock equations are derived the same way as normal shock equations, but applied only to the velocity component normal to the shock wave. The tangential velocity component is unchanged across the shock.

This means you can use the normal shock relations with the normal Mach number $M_{n1} = M_1 \sin\beta$ substituted for $M_1$. The downstream normal Mach number $M_{n2}$ then gives you the full downstream Mach number through the geometry.

Deflection angle vs shock angle

For a given upstream Mach number, the relationship between $\theta$ and $\beta$ is described by the $\theta$-$\beta$-$M$ relation. Key features of this relationship:

• For each $\theta$, there are generally two possible shock angles: a weak shock (smaller $\beta$, supersonic downstream flow) and a strong shock (larger $\beta$, often subsonic downstream flow). In practice, the weak solution usually occurs.
• There's a maximum deflection angle for each $M_1$. If the required deflection exceeds this maximum, the oblique shock detaches from the body and becomes a curved, detached bow shock.
• At $\theta = 0°$, the weak solution gives a Mach wave ($\beta = \mu$), and the strong solution gives a normal shock ($\beta = 90°$).

Mach number relations across oblique shocks

The downstream Mach number depends on $M_1$, $\beta$, and $\gamma$. As the shock angle increases toward 90°, the oblique shock approaches a normal shock and the downstream Mach number decreases. For weak oblique shocks (small $\beta$), the downstream flow remains supersonic.

Pressure ratio across oblique shocks

The pressure ratio across an oblique shock depends on $M_1$ and $\beta$:

• It's always greater than 1 (pressure increases).
• It increases as $\beta$ increases (stronger shock).
• For a Mach wave ($\beta = \mu$), the pressure ratio is exactly 1 (no change).

Density ratio across oblique shocks

The density ratio follows the same trends as the pressure ratio:

• Always greater than 1.
• Increases with increasing shock angle.
• Bounded by the same limit of $\frac{\gamma+1}{\gamma-1}$ as for normal shocks.

Prandtl-Meyer expansion waves

Prandtl-Meyer expansion waves form when supersonic flow turns around a sharp convex corner or encounters a smooth outward expansion. They're the isentropic counterpart to shock waves: where shocks compress and decelerate the flow, expansion waves expand and accelerate it.

Because the process is isentropic, there are no losses in total pressure. This makes expansion waves "free" in terms of efficiency, which is why supersonic vehicle designers prefer gradual expansions over abrupt compressions.

Prandtl-Meyer function definition

The Prandtl-Meyer function $\nu(M)$ gives the total turning angle associated with an isentropic expansion from Mach 1 to Mach $M$:

$\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}}\tan^{-1}\sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)}-\tan^{-1}\sqrt{M^2-1}$

At $M = 1$, $\nu = 0°$. For air ($\gamma = 1.4$), the maximum value of $\nu$ as $M \to \infty$ is about 130.45°, which represents the maximum possible turning angle through an isentropic expansion.

Mach number vs Prandtl-Meyer function

The Prandtl-Meyer function is monotonically increasing with Mach number. To use it for a problem:

1. Look up (or calculate) $\nu_1$ for the upstream Mach number $M_1$.
2. Add the turning angle $\theta$: $\nu_2 = \nu_1 + \theta$.
3. Invert the Prandtl-Meyer function to find $M_2$ from $\nu_2$ (this typically requires a table or numerical solver).

The downstream Mach number is always higher than the upstream Mach number.

Expansion wave geometry

Expansion waves are fans of Mach waves centered at the corner. Each Mach wave makes the Mach angle with the local flow direction:

$\mu = \sin^{-1}\left(\frac{1}{M}\right)$

At the start of the fan (upstream), the Mach angle corresponds to $M_1$. At the end (downstream), it corresponds to $M_2$. Since $M_2 > M_1$, the downstream Mach angle is smaller, and the fan spreads out from the corner.

Expansion wave equations

Because the process is isentropic, you can use the isentropic flow relations to find downstream properties. For a calorically perfect gas:

$\frac{p_2}{p_1} = \left(\frac{1+\frac{\gamma-1}{2}M_1^2}{1+\frac{\gamma-1}{2}M_2^2}\right)^{\frac{\gamma}{\gamma-1}}$

$\frac{T_2}{T_1} = \frac{1+\frac{\gamma-1}{2}M_1^2}{1+\frac{\gamma-1}{2}M_2^2}$

$\frac{\rho_2}{\rho_1} = \left(\frac{1+\frac{\gamma-1}{2}M_1^2}{1+\frac{\gamma-1}{2}M_2^2}\right)^{\frac{1}{\gamma-1}}$

These are the standard isentropic relations, applied once you know $M_2$ from the Prandtl-Meyer function.

Mach number relations across expansion waves

The upstream and downstream Mach numbers are linked through the Prandtl-Meyer function:

$\nu(M_2) = \nu(M_1) + \theta$

The downstream Mach number is always greater than the upstream value. The amount of acceleration depends on both the turning angle $\theta$ and $\gamma$.

Pressure ratio across expansion waves

The pressure ratio is always less than 1 (pressure drops). A larger turning angle produces a greater pressure drop and stronger expansion. You calculate it using the isentropic relation above once $M_2$ is known.

Density ratio across expansion waves

The density ratio is also always less than 1 (density drops). It follows the same isentropic relation and trends as the pressure ratio: larger turning angles produce greater decreases.

Supersonic nozzle flow

Converging-diverging (CD) nozzles are the primary device for accelerating flow from subsonic to supersonic speeds. They're used in rocket engines, jet engine exhaust nozzles, and supersonic wind tunnels. The physics of how they work ties together isentropic flow, choking, and shock wave formation.

Converging-diverging nozzle geometry

A CD nozzle has three sections:

• Converging section: Flow accelerates from subsonic toward sonic speed.
• Throat: The minimum cross-sectional area, where $M = 1$ when the flow is choked.
• Diverging section: Flow can either decelerate (subsonic) or accelerate further (supersonic), depending on the back pressure.

The area ratio $A/A^*$ (exit area to throat area) determines the exit Mach number for fully isentropic flow. Each area ratio corresponds to two possible Mach numbers: one subsonic and one supersonic.

Isentropic flow in nozzles

Isentropic flow means no heat transfer (adiabatic) and no irreversibilities (no shocks, no friction). In an ideal CD nozzle, the isentropic relations connect Mach number to the ratios of local properties to stagnation (total) properties:

$\frac{T}{T_0} = \left(1+\frac{\gamma-1}{2}M^2\right)^{-1}$

$\frac{p}{p_0} = \left(1+\frac{\gamma-1}{2}M^2\right)^{-\frac{\gamma}{\gamma-1}}$

$\frac{\rho}{\rho_0} = \left(1+\frac{\gamma-1}{2}M^2\right)^{-\frac{1}{\gamma-1}}$

These let you find pressure, temperature, and density at any point in the nozzle if you know the local Mach number.

Choked flow conditions

Choking occurs when the Mach number at the throat reaches exactly 1. Once choked:

• The mass flow rate is at its maximum and no longer increases if you lower the back pressure further.
• The mass flow rate depends only on the stagnation conditions ($p_0$, $T_0$) and the throat area.

The critical pressure ratio at the throat for choking is:

$\frac{p^*}{p_0} = \left(\frac{2}{\gamma+1}\right)^{\frac{\gamma}{\gamma-1}}$

For air ($\gamma = 1.4$), this gives $p^*/p_0 \approx 0.528$. The throat pressure must drop to about 52.8% of the stagnation pressure for the flow to choke.

Nozzle flow regimes

Depending on the ratio of back pressure ($p_b$) to stagnation pressure ($p_0$), several flow regimes exist:

• Fully subsonic: $p_b/p_0$ is high. Flow accelerates in the converging section and decelerates in the diverging section. Never reaches $M = 1$.
• Choked, subsonic exit: $M = 1$ at the throat, but the diverging section acts as a diffuser. Flow exits subsonically.
• Isentropic supersonic exit: $p_b$ exactly matches the supersonic isentropic exit pressure. Flow is supersonic throughout the diverging section with no shocks.
• Over-expanded: $p_b$ is higher than the isentropic supersonic exit pressure but lower than the subsonic exit pressure. A shock forms somewhere in the diverging section.
• Under-expanded: $p_b$ is lower than the isentropic supersonic exit pressure. Expansion waves form at the nozzle exit as the flow continues to expand outside the nozzle.

Over-expanded vs under-expanded nozzles

• Over-expanded: The nozzle exit pressure is lower than the back pressure. The flow "over-expands" and must compress back through shock waves inside or just outside the nozzle. This reduces thrust and efficiency.
• Under-expanded: The nozzle exit pressure is higher than the back pressure. The flow hasn't expanded enough inside the nozzle, so expansion waves form at the exit. Some potential thrust is lost because the expansion happens outside the nozzle where it can't push on the nozzle walls.
• Perfectly expanded (optimal): Exit pressure equals back pressure. No shocks or expansion waves at the exit. Maximum thrust for the given conditions.

Shock waves in nozzles

When a normal shock forms in the diverging section of an over-expanded nozzle:

1. The flow is supersonic upstream of the shock.
2. The shock causes an abrupt jump to subsonic conditions.
3. The flow then decelerates subsonically through the rest of the diverging section.
4. The exit pressure adjusts to match the back pressure.

As the back pressure decreases, the shock moves downstream toward the exit. If the back pressure drops enough, the shock reaches the exit plane and eventually moves outside the nozzle entirely. Shocks inside nozzles cause total pressure losses and reduce thrust, so proper nozzle design aims to avoid them during normal operation.

Supersonic airfoil theory

Supersonic airfoil theory describes how airfoils generate lift and drag when the freestream Mach number exceeds 1. The flow physics are fundamentally different from subsonic aerodynamics: instead of smooth pressure distributions governed by circulation, supersonic airfoils rely on shock and expansion wave patterns to create pressure differences between upper and lower surfaces.